
theorem Th50: :: Mycth:
for n being Nat
 holds clique# Mycielskian n = 2 & chromatic# Mycielskian n = n+2
proof
  defpred P[Nat] means
   clique# Mycielskian $1 = 2 & chromatic# Mycielskian $1 = $1+2;
 A1: clique# Mycielskian 0 = clique# CompleteRelStr 2 by Th49
     .= 2 by Th33;
   chromatic# Mycielskian 0 = chromatic# CompleteRelStr 2 by Th49
     .= 2 by Th35;
     then
 A2: P[0] by A1;
 A3: for n being Nat st P[n] holds P[n+1] proof
      let n be Nat such that
     A4: clique# Mycielskian n = 2 and
     A5: chromatic# Mycielskian n = n+2;
     A6: Mycielskian (n+1) = Mycielskian Mycielskian n by Th49;
      thus clique# Mycielskian (n+1) = 2 by A4,A6,Th46;
      thus chromatic# Mycielskian (n+1)
         = 1 + chromatic# Mycielskian n by A6,Th48
        .= n+1+2 by A5;
     end;
 for n being Nat holds P[n] from NAT_1:sch 2(A2, A3);
 hence thesis;
end;
